Correlation between superfluid density and Tc of underdoped YBa2Cu3O6+x near the superconductor-insulator transition

We report measurements of the ab-plane superfluid density Ns (magnetic penetration depth, \lambda) of severely underdoped films of YBa2Cu3O6+x, with Tc's from 6 to 50 K. Tc is not proportional to Ns(0); instead, we find Tc ~ Ns^{1/2.3 +/- 0.4}. At the lowest dopings, Tc is as much as 5 times larger than the upper limit set by the KTB transition temperature of individual CuO2 bilayers.